London Tourist Sites

Changing of the guard is free, so just walk up to the palace gates. But everyone else will be doing the same, so the earliest arrivals get the choice of the best viewing spots. October probably won’t be as busy as dinner, though, so simply showing up won’t be so challenging.

Churchill War Rooms and Imperial War Museum are 2 different museums, at 2 different locations. The Churchill Rooms have the WW II underground bunker plus a section devoted to the life of Winston Churchill. There’s an admission fee and advance tickets would be wise, maybe even required nowadays.

Imperial War Museum is free, but back in May there was a table at the entrance where employees urged visitors to contribute £5 and get the museum guidebook in the bargain. No timed entry required. There are vehicles and planes hanging in the center space, with separate wings for WW I, WW II, and the Holocaust. A special, smaller temporary exhibit about The Troubles in Northern Ireland was there in May.

I'd recommend the Tower of London as soon as it opens. Head straight to the Crown Jewels, or just before and to the right you can get the obligatory photo of a red coated guard in his furry hat, like you see inside the gates of Buckingham Palace. We did get a photo, but 15 minutes later after the crown jewels, the poor guy was mobbed with people wanting a photo.

You don't have Hampton Court Palace on your list. I'd rank that almost as high as the Tower. We spent about 6 hours there.

Reserve Churchill War Rooms, Buckingham Palace, Westminster Abbey and Tower of London. Book
as early in the day as you can so you will be able to see other things. Opening hours are most likely 10am but research to be certain!

Wicked has been playing for years. Not sure if its necessary to pre book. I’d walk up to the ticket booth for same day tickets but thats just me. If an absolute must see, pre book. It is well done.

The Imperial War Museum is not the Churchill War Rooms.

Westminster Abbey, the Tower of London, and Churchill War Rooms will likely have long lines for admission, even in October, so I would get tickets in advance for those. And to have a better experience, with reduced crowd level at Westminster Abbey and the Tower of London, I would book the first available slot in the morning.

If by “Buckingham Palace” you mean to actually go inside, that will not be possible in October. Visits to the State Rooms and tours of the palace are only offered through September 24. But as noted there is no fee (and no crowd control) for watching the Changing of the Guard.

https://www.rct.uk/visit/buckingham-palace

We prefer the guard change ceremony at the Horse Guards Parade by St. James Park. It is much less crowded (or at least it was when we last watched).

https://changing-guard.com/Kings-Life-Guard.html

The British Museum is free (donations encouraged) and always crowded. The Egyptian rooms and area of the Elgin Marbles tend to be the most crowded. We found peace and quiet and art to ourselves downstairs in the African Art section last time we were there.

For theatre tickets, since you have a specific show in mind, I personally would buy tickets to the performance you want now for best selection of seats and prices. We have never waited for the reduced-price tickets on sale the day of performance at one of the cheap ticket booths; for the shows we have seen I doubt there are any tickets left to go on sale.

It appears the Apollo Theatre is nice and open, not cramped and steep like some London theatres. You can see reviews of the various seats on sites such as this:

https://seatplan.com/london/apollo-victoria-theatre/

WA - if you want an early slot, book - that holds all year.

CTG is a street event, no booking required.

BP will not be fully open - I would be surprised if the open sections booked up in October.

Tower - I would always go first thing for the Crown Jewels but there shouldn't be timed slots. But easier to get tickets in advance.

CWR - you only tend to see a queue in the summer.

IWM - walk up

BM - walk up

Theatre - book so you get seats you want at a price you want. The much vaunted reduced price tickets are often just expensive ones and you can find there are just as goods that are cheaper at full price.

My personal rule is if I really want to go, book.